Rota-Baxter operators were first appeared in a 1960's paper of Baxter [1]} as a tool to study fluctuation theory in probability. Subsequently, such operators were investigated by Rota [2]}, Cartier [3]}, Atkinson [4]} among others. In the last twenty years, Rota-Baxter operators received very much attention due to their connection with many branches of mathematics and mathematical physics. For instance, Rota-Baxter operators play a key role in the combinatorial study of trees and shuffles [5]}, splitting of algebras [6]}, infinitesimal bialgebras [7]}, Hopf algebras [8]}, double algebras [9]} and renormalizations in quantum field theory [10]}.
Let \(A\)  be an associative algebra. A linear map \(R: A \rightarrow A\)  is said to be a Rota-Baxter operator on \(A\)  if
\(R(a) \cdot R(a^{\prime }) = R \big ( R(a) \cdot a^{\prime } ~+~ a \cdot R(a^{\prime }) \big ), \text{ for } a, a^{\prime } \in A.\) 
